A280879 Occurrences of decrease of the probability density P(n) of coprime numbers k,m, satisfying 1 <= k <= a(n) and 1 <= m <= a(n), and a(n) congruent to 1 (mod 2) and a(n) not congruent to 3 (mod 6).

5005, 6545, 7315, 7735, 8645, 8855, 10465, 11165, 11935, 14245, 25025, 32725, 35035, 36575, 38675, 43225, 44275, 45815, 51205, 52325, 54145, 55055, 55825, 59675, 60515, 61985, 65065, 71225, 71995, 73255, 78155, 80465, 83545, 85085, 95095, 97405, 99715

Offset

1,1

Comments

Probability densities satisfying P(a(n)) < P(a(n)-1) and a(n) congruent to 1 (mod 2) and a(n) not congruent to 3 (mod 6).

It appears that most numbers satisfy a(n) congruent to 35 (mod 70), but a(74) congruent to 15 (mod 70) and a(93) congruent to 55 (mod 70).

Subset of A280877.

P(n) = ((2*Sum_{m=1..a(n)} phi(m))-1)/a(n)^2 (Cf. Euler phi function A000010).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..102 from A.H.M. Smeets)

Prog

(Python)
from fraction import gcd
t = 1
to = 1
i = 1
x = 1
while x > 0:
....x = x + 1
....y = 0
....while y < x:
........y = y + 1
........if gcd(x, y) == 1:
............t = t + 2
....e = t*(x-1)*(x-1) - to*x*x
....if (e < 0 and x%2 == 1 and x%6 != 3):
........print(i, x)
........i = i + 1
....to = t
(PARI) P(n) = (2 *sum(j=1, n, eulerphi(j)) - 1)/n^2;
isok(n) = (n % 2) && ((n % 6) != 3) && (P(n) < P(n-1)); \\ Michel Marcus, Jan 29 2017
(Python)
from sympy import totient
A280879_list, n, t = [], 1, 1
while len(A280879_list) < 1000:
    n += 1
    h = totient(n)
    t2 = t+h
    if n % 2 and n % 6 != 3 and 2*(n*(h*n - 2*t2 + 1) + t2) <  1:
        A280879_list.append(n)
    t = t2 # Chai Wah Wu, Feb 11 2017

Crossrefs

Cf. A018805, A280877, A280878.

Sequence in context: A035787 A108011 A018188 * A067226 A249081 A154059

Adjacent sequences: A280876 A280877 A280878 * A280880 A280881 A280882

Keyword

nonn

Author

A.H.M. Smeets, Jan 09 2017

Status

approved